While the Ollivier curvature of a graph is based on a linear optimization problem, it turns out to be surprisingly fruitful to investigate the curvature via non-linear evolution equations. Specifically, we show how a discrete Ricci flow version and the search for Busemann-type functions can be unified in an abstract setting of non-linear Markov chains. In this abstract setting, we show long term convergence which we then apply to prove convergence of the discrete Ricci flow, as well as the existence of a Busemann type function as the limit of a certain non-linear Markov chain which we call Laplacian separation flow.